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doi: 10.3934/eect.2021008
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## Almost periodic type functions and densities

* Corresponding author: Marko Kostić

Received  July 2020 Early access January 2021

Fund Project: The author is supported by grant no. 451-03-68/2020/14/200156, Ministry of Science and Technological Development, Republic of Serbia

In this paper, we introduce and analyze the notions of $\odot_{g}$-almost periodicity and Stepanov $\odot_{g}$-almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) $g$-densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems.

Citation: Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, doi: 10.3934/eect.2021008
##### References:
 [1] S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer), 6 (2012), 5 pp. doi: 10.1186/2251-7456-6-29.  Google Scholar [2] B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp.  Google Scholar [3] B. Basit and H. Güenzler, On spectral criteria for solutions of evolution equations and comments on reduced spectra, Far East J. Math. Sci. (FJMS), 65 (2012), 273-288.   Google Scholar [4] M. V. Bebutov, On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.   Google Scholar [5] A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar [6] H. Bohr, Zur theorie der fastperiodischen Funktionen Ⅰ; Ⅱ; Ⅲ, Acta Math., 45 (1924), 29–127; H6 (1925), 101–214; H5 (1926), 237–281. doi: 10.1007/BF02395468.  Google Scholar [7] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar [8] L. I. Danilov, The uniform approximation of recurrent functions and almost recurrent functions, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 4 (2013), 36-54.   Google Scholar [9] J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar [10] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar [11] H.-S. Ding, J. Liang and T.-J. Xiao, Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal., 88 (2009), 1079-1091.  doi: 10.1080/00036810903156164.  Google Scholar [12] H.-S. Ding, W. Long and G. M. N'Guérékata, Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients, J. Comput. Anal. Appl., 13 (2011), 231-242.   Google Scholar [13] H.-S. Ding and S.-M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar [14] T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Text in Mathematics, vol. 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar [15] K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000.  Google Scholar [16] A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar [17] A. M. Fink, Extensions of almost automorphic sequences, J. Math. Anal. Appl., 27 (1969), 519-523.  doi: 10.1016/0022-247X(69)90132-2.  Google Scholar [18] A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp.  Google Scholar [19] A. Geroldinger and I. Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8962-8.  Google Scholar [20] G. Grekos, V. Toma and J. Tomanová, A note on uniform or Banach density, Ann. Math. Blaise Pascal, 17 (2010), 153-163.  doi: 10.5802/ambp.280.  Google Scholar [21] G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar [22] G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005.  Google Scholar [23] G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar [24] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations, 49 (1983), 473-483.  doi: 10.1016/0022-0396(83)90008-6.  Google Scholar [25] A. Haraux and P. Souplet, An example of uniformly recurrent function which is not almost periodic, J. Fourier Anal. Appl., 10 (2004), 217-220.  doi: 10.1007/s00041-004-8012-4.  Google Scholar [26] H. R. Henríquez, On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl., 146 (1990), 420-433.   Google Scholar [27] E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948.  Google Scholar [28] D. Ji and Y. Lu, Stepanov-like pseudo almost automorphic solution to a parabolic evolution equation, Adv. Difference Equ., 341 (2015), 17 pp. doi: 10.1186/s13662-015-0667-4.  Google Scholar [29] M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019. doi: 10.1515/9783110641851.  Google Scholar [30] M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. Google Scholar [31] M. Kostić, ${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.  doi: 10.2298/pim1920001k.  Google Scholar [32] M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020). doi: 10.1007/s00574-020-00197-7.  Google Scholar [33] B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions]  Google Scholar [34] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.   Google Scholar [35] P. Ribenboim, Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.   Google Scholar [36] A. M. Samoilenko and S. I. Trofimchuk, Unbounded functions with almost periodic differences, Ukrainian Math. J., 43 (1991), 1306-1309.  doi: 10.1007/BF01061818.  Google Scholar [37] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar [38] W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.  Google Scholar [39] R. Xie and C. Zhang, Space of $\omega$-periodic limit functions and its applications to an abstract Cauchy problem, J. Function Spaces, vol. 2015, Art. ID 953540, 10 pp. doi: 10.1155/2015/953540.  Google Scholar [40] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985.  Google Scholar [41] C. Zhang, Ergodicity and asymptotically almost periodic solutions of some differential equations, Int. J. Math. Math. Sci., 25 (2001), 787-800.  doi: 10.1155/S016117120100429X.  Google Scholar [42] H. Y. Zhao and M. Fečkan, Pseudo almost periodic solutions of an iterative equation with variable coefficients, Miskolc Math. Notes, 18 (2017), 515-524.  doi: 10.18514/MMN.2017.2047.  Google Scholar

show all references

##### References:
 [1] S. Abbas, A note on Weyl pseudo almost automorphic functions and their properties, Math. Sci. (Springer), 6 (2012), 5 pp. doi: 10.1186/2251-7456-6-29.  Google Scholar [2] B. Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math., 338 (1995), 26 pp.  Google Scholar [3] B. Basit and H. Güenzler, On spectral criteria for solutions of evolution equations and comments on reduced spectra, Far East J. Math. Sci. (FJMS), 65 (2012), 273-288.   Google Scholar [4] M. V. Bebutov, On dynamical systems in the space of continuous functions, Byull. Moskov. Gos. Univ. Mat., 2 (1940), 1-52.   Google Scholar [5] A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar [6] H. Bohr, Zur theorie der fastperiodischen Funktionen Ⅰ; Ⅱ; Ⅲ, Acta Math., 45 (1924), 29–127; H6 (1925), 101–214; H5 (1926), 237–281. doi: 10.1007/BF02395468.  Google Scholar [7] H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar [8] L. I. Danilov, The uniform approximation of recurrent functions and almost recurrent functions, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 4 (2013), 36-54.   Google Scholar [9] J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar [10] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar [11] H.-S. Ding, J. Liang and T.-J. Xiao, Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Appl. Anal., 88 (2009), 1079-1091.  doi: 10.1080/00036810903156164.  Google Scholar [12] H.-S. Ding, W. Long and G. M. N'Guérékata, Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients, J. Comput. Anal. Appl., 13 (2011), 231-242.   Google Scholar [13] H.-S. Ding and S.-M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar [14] T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Graduate Text in Mathematics, vol. 272, Springer, Cham, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar [15] K.-J. Engel and R. Nagel, One–Parameter Semigroups for Linear Evolution Equations, Springer–Verlag, New York, 2000.  Google Scholar [16] A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin-New York, 1974.  Google Scholar [17] A. M. Fink, Extensions of almost automorphic sequences, J. Math. Anal. Appl., 27 (1969), 519-523.  doi: 10.1016/0022-247X(69)90132-2.  Google Scholar [18] A. M. Fink, Almost Periodic Points in Topological Transformation Semi-groups, Ph.D thesis, Iowa State University, Digital Repository (1960), 44 pp.  Google Scholar [19] A. Geroldinger and I. Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8962-8.  Google Scholar [20] G. Grekos, V. Toma and J. Tomanová, A note on uniform or Banach density, Ann. Math. Blaise Pascal, 17 (2010), 153-163.  doi: 10.5802/ambp.280.  Google Scholar [21] G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar [22] G. M. N'Guérékata, Topics in Almost Automorphy, Springer–Verlag, New York, 2005.  Google Scholar [23] G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar [24] A. Haraux, Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes, J. Differential Equations, 49 (1983), 473-483.  doi: 10.1016/0022-0396(83)90008-6.  Google Scholar [25] A. Haraux and P. Souplet, An example of uniformly recurrent function which is not almost periodic, J. Fourier Anal. Appl., 10 (2004), 217-220.  doi: 10.1007/s00041-004-8012-4.  Google Scholar [26] H. R. Henríquez, On Stepanov-almost periodic semigroups and cosine functions of operators, J. Math. Anal. Appl., 146 (1990), 420-433.   Google Scholar [27] E. Hille, Functional Analysis and Semi-Groups, American Mathematical Society, New York, 1948.  Google Scholar [28] D. Ji and Y. Lu, Stepanov-like pseudo almost automorphic solution to a parabolic evolution equation, Adv. Difference Equ., 341 (2015), 17 pp. doi: 10.1186/s13662-015-0667-4.  Google Scholar [29] M. Kostić, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, De Gruyter, Berlin, 2019. doi: 10.1515/9783110641851.  Google Scholar [30] M. Kostić, Chaos for Linear Operators and Abstract Differential Equations, Nova Science Publishers Inc., New York, 2020. Google Scholar [31] M. Kostić, ${\mathcal F}$-Hypercyclic operators on Fréchet spaces, Publ. Inst. Math. (Beograd) (N.S.), 106 (2019), 1-18.  doi: 10.2298/pim1920001k.  Google Scholar [32] M. Kostić, Quasi-asymptotically almost periodic functions and applications, Bull. Braz. Math. Soc., New Series, (2020). doi: 10.1007/s00574-020-00197-7.  Google Scholar [33] B. M. Levitan, Počti-periodičeskie funkcii, (Russian) [Almost Periodic Functions]  Google Scholar [34] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, London, 1982.   Google Scholar [35] P. Ribenboim, Density results on families of Diophantine equations with finitely many solutions, Enseign. Math. (2), 39 (1993), 3-23.   Google Scholar [36] A. M. Samoilenko and S. I. Trofimchuk, Unbounded functions with almost periodic differences, Ukrainian Math. J., 43 (1991), 1306-1309.  doi: 10.1007/BF01061818.  Google Scholar [37] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar [38] W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.  Google Scholar [39] R. Xie and C. Zhang, Space of $\omega$-periodic limit functions and its applications to an abstract Cauchy problem, J. Function Spaces, vol. 2015, Art. ID 953540, 10 pp. doi: 10.1155/2015/953540.  Google Scholar [40] S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Math., vol.126, Pitman, Boston, MA, 1985.  Google Scholar [41] C. Zhang, Ergodicity and asymptotically almost periodic solutions of some differential equations, Int. J. Math. Math. Sci., 25 (2001), 787-800.  doi: 10.1155/S016117120100429X.  Google Scholar [42] H. Y. Zhao and M. Fečkan, Pseudo almost periodic solutions of an iterative equation with variable coefficients, Miskolc Math. Notes, 18 (2017), 515-524.  doi: 10.18514/MMN.2017.2047.  Google Scholar
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